There are many different strategies available for constructing prediction intervals, one of the most common is Bayesian, which is based on a fully-specified probability model for the observables. In its most general form, the Bayesian approach starts with a prior distribution for \(\mathsf {P}\) , which determines a prior predictive distribution and, ultimately, a posterior predictive distribution for \(Y_{n+1}\) , given \(Y^n=y^n\) . Do prediction intervals derived from a Bayesian predictive distribution satisfy (REF )? To investigate this, we consider a simple-yet-powerful Bayesian nonparametric approach, one that assumes data are iid with common marginal distribution, to which we assign a Dirichlet process prior [1]} with base measure \(G\) —a probability measure on the \(Y\)  space—and precision parameter \(\delta > 0\) . This model is rather flexible and has strong theoretical support; see, e.g., [2]} and [3]}. With this formulation, it turns out that the posterior predictive distribution, \(\Pi _{Y^n}\) , for \(Y_{n+1}\)  is quite simple: it is a mixture of \(G\)  and the empirical distribution \(\mathbb {P}_n\)  based on \(Y^n\) , i.e.,
\( \Pi _{Y^n}(A) = \tfrac{1}{\delta + n} \, G(A) + (1-\tfrac{1}{\delta + n}) \, \mathbb {P}_n(A), \quad A \subseteq \mathbb {R}. \) 
